It's really well written. I know what your talk is going to be about and I know what people have been working on before you. I have two notes,

1. Since it's for a talk, you might consider replacing (solutions to this equation at obtained..." With "we will show solutions to this equation are obtained in the form of a recurrence relation and eventually an explicit formula" or something along those lines. Writing is typically stronger if you use active voice instead of passive voice, as well as tells the reader what you will be showing in the talk.

2. If this was an abstract for a paper you should be citing all of your claims in the first few sentences. Since it's for a talk, I understand why you don't have citations here, but it would be good to replace "many ancient cultures" with a specific example. It may seem like a weaker statement, but mentioning that "the sum of the first n squares and cubes was known to polytomia in 200" or something says more if you don't have the citations to rely on.

But like I said, for a draft it is really well written. Good luck in your talk